The Gaussian propeller conjecture in every dimension

1 · The essentials

Imagine dividing ordinary space into finitely many regions and giving points near the origin more weight than points far away, according to a Gaussian bell curve. For each region, add up its position arrows using those weights, without dividing by the region’s total weight. Square the length of each resulting arrow and add the answers. The main result says this total can never exceed $9/ (8 π)$ , regardless of the dimension or the number of regions. In two or more dimensions, three equal wedges meeting at the origin reach this limit; in higher dimensions, they extend freely in the remaining directions. The main Gaussian bound has a Lean formalization, providing computer-checked verification.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a circular tray covered with sand that is densest near its center, with each grain contributing a pull in the direction of its position. Assign the grains to different teams, combine each team’s pulls, and score the squared strength of each combined pull. The geometric result limits the total score, with three equal wedges achieving the limit in the stated setting.

Why it matters

The result gives one sharp upper bound that works in every dimension, with a simple configuration attaining it in dimensions at least two. The repository also reports a computational consequence: combined with a separate Unique Games theorem, it yields an NP-hardness statement for improving a specified loss factor in identity-target kernel clustering. That consequence uses an additional theorem, not just the geometric bound.

2 · Going further

Let $A_{1}, \dots, A_{m}$ be a finite measurable partition of a Euclidean space, and let $γ$ denote its Gaussian measure. The Gaussian first moment of a cell is the vector $v_{i} = \int_{A_{i}} x d γ (x)$ ; this is not divided by the Gaussian measure of the cell. The precise bound is $\sum_{i = 1}^{m} ∥ v_{i} ∥^{2} \leq 9/ (8 π)$ . The supplied manuscript identifies this statement as the Gaussian propeller conjecture, and the main bound has a Lean formalization.

For dimension at least two and at least three cells, the manuscript states that the bound is attained by three planar sectors of angle $2 π /3$ , extended by an orthogonal Euclidean factor. Thus the attaining construction is planar in its essential geometry, even when the ambient space has more dimensions. The supplied abstract states the bound and its attainment but does not describe the argument establishing the upper bound, so no proof mechanism can be extracted from this source.

The repository summary links this geometric result to a separate Unique Games theorem. It reports that their combination establishes NP-hardness of improving the loss factor $(8 π /9) (1 - 1/ k)$ for identity-target kernel clustering, with fixed $k \geq 3$ , on rational centered positive semidefinite inputs. No companion abstract or argument is supplied, and the stated Lean verification should not be extended to that computational consequence.

Useful notions

Vectors and squared lengths
Gaussian weighting
Measurable partitions
Integration of vector-valued quantities
Orthogonal directions

What was known before / what is new

Before

The source names the geometric statement the Gaussian propeller conjecture. It also refers to a separate Unique Games theorem used for the computational consequence, but gives no account of earlier partial results or historical development.

New

The main manuscript establishes the Gaussian propeller bound for every finite measurable partition in every dimension, and the main result has a Lean formalization. It also states attainment by three equal planar sectors, extended orthogonally, when the dimension is at least two and at least three cells are available. The repository reports the kernel-clustering hardness consequence obtained by combining this result with the separate Unique Games theorem.

Limits and assumptions

The upper bound concerns finite measurable partitions and Gaussian first moments, not arbitrary weighting rules. The stated attaining construction requires dimension at least two and at least three cells; the source does not give an attainment description outside those conditions or classify all equality cases. The main Gaussian result has a Lean formalization, but the supplied material does not specify its detailed coverage of every associated claim. In particular, it does not state that the separate Unique Games theorem or the clustering consequence is formalized. The computational statement is restricted to fixed $k \geq 3$ and rational centered positive semidefinite inputs. No further open questions are identified in the supplied text.

Mini glossary

Measurable partition: A division of a space into disjoint cells covering the space, with each cell measurable under the relevant measure.
Gaussian first moment: The vector obtained by integrating positions over a region using Gaussian weights, without normalizing by the region’s measure.
Planar sector: A wedge-shaped region in a plane between two rays from a common point.
Orthogonal Euclidean factor: Additional perpendicular coordinate directions along which a planar construction is extended without restriction.
Lean formalization: A mathematical statement and its supporting proof expressed for checking by the Lean proof assistant.

Keywords : Gaussian propeller conjecture · Gaussian first moments · Kernel clustering

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $9/ (8 π)$ . In dimension at least two, three planar sectors of angle $2 π /3$ , extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $(8 π /9) (1 - 1/ k)$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

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lean/docs/096.md

Manuscripts in this family

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The Gaussian propeller bound in every dimension

24/09/2026

Verified by Lean

We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most $9/ (8 π)$ . For dimension at least two and at least three cells, three planar sectors of angle $2 π /3$ , extended by an orthogonal Euclidean factor, attain the bound.

PDF GitHub folder Lean formalization